cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A344296 Numbers with at least as many prime factors (counted with multiplicity) as half their sum of prime indices.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, 24, 27, 28, 30, 32, 36, 40, 48, 54, 56, 60, 64, 72, 80, 81, 84, 88, 90, 96, 100, 108, 112, 120, 128, 144, 160, 162, 168, 176, 180, 192, 200, 208, 216, 224, 240, 243, 252, 256, 264, 270, 280, 288, 300, 320, 324, 336, 352
Offset: 1

Views

Author

Gus Wiseman, May 16 2021

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
These are the Heinz numbers of certain partitions counted by A025065, but different from palindromic partitions, which have Heinz numbers A265640.

Examples

			The sequence of terms together with their prime indices begins:
      1: {}            30: {1,2,3}
      2: {1}           32: {1,1,1,1,1}
      3: {2}           36: {1,1,2,2}
      4: {1,1}         40: {1,1,1,3}
      6: {1,2}         48: {1,1,1,1,2}
      8: {1,1,1}       54: {1,2,2,2}
      9: {2,2}         56: {1,1,1,4}
     10: {1,3}         60: {1,1,2,3}
     12: {1,1,2}       64: {1,1,1,1,1,1}
     16: {1,1,1,1}     72: {1,1,1,2,2}
     18: {1,2,2}       80: {1,1,1,1,3}
     20: {1,1,3}       81: {2,2,2,2}
     24: {1,1,1,2}     84: {1,1,2,4}
     27: {2,2,2}       88: {1,1,1,5}
     28: {1,1,4}       90: {1,2,2,3}

Crossrefs

The case with difference at least 1 is A322136.
The case of equality is A340387, counted by A000041 or A035363.
The opposite version is A344291, counted by A110618.
The conjugate version is A344414, with even-weight case A344416.
A025065 counts palindromic partitions, ranked by A265640.
A056239 adds up prime indices, row sums of A112798.
A300061 lists numbers whose sum of prime indices is even.
Cf. A001399, A002865, A025147, A027336, A036036, A067712, A244990, A261144, A325691, A344293, A344295.

Programs

  • Mathematica
    Select[Range[100],PrimeOmega[#]>=Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]/2&]

Formula

A056239(a(n)) <= 2*A001222(a(n)).
a(n) = A322136(n)/4.

A344293 5-smooth numbers n whose sum of prime indices A056239(n) is at least twice the number of prime indices A001222(n).

Original entry on oeis.org

1, 3, 5, 9, 10, 15, 25, 27, 30, 45, 50, 75, 81, 90, 100, 125, 135, 150, 225, 243, 250, 270, 300, 375, 405, 450, 500, 625, 675, 729, 750, 810, 900, 1000, 1125, 1215, 1250, 1350, 1500, 1875, 2025, 2187, 2250, 2430, 2500, 2700, 3000, 3125, 3375, 3645, 3750, 4050
Offset: 1

Views

Author

Gus Wiseman, May 16 2021

Keywords

Comments

A number is 5-smooth if its prime divisors are all <= 5.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The sequence of terms together with their prime indices begins:
       1: {}            125: {3,3,3}
       3: {2}           135: {2,2,2,3}
       5: {3}           150: {1,2,3,3}
       9: {2,2}         225: {2,2,3,3}
      10: {1,3}         243: {2,2,2,2,2}
      15: {2,3}         250: {1,3,3,3}
      25: {3,3}         270: {1,2,2,2,3}
      27: {2,2,2}       300: {1,1,2,3,3}
      30: {1,2,3}       375: {2,3,3,3}
      45: {2,2,3}       405: {2,2,2,2,3}
      50: {1,3,3}       450: {1,2,2,3,3}
      75: {2,3,3}       500: {1,1,3,3,3}
      81: {2,2,2,2}     625: {3,3,3,3}
      90: {1,2,2,3}     675: {2,2,2,3,3}
     100: {1,1,3,3}     729: {2,2,2,2,2,2}

Crossrefs

Allowing any number of parts and sum gives A051037, counted by A001399.
These are Heinz numbers of the partitions counted by A266755.
Allowing parts > 5 gives A344291, counted by A110618.
The non-3-smooth case is A344294, counted by A325691.
Requiring the sum of prime indices to be even gives A344295.
A000070 counts non-multigraphical partitions, ranked by A344292.
A025065 counts partitions of n with >= n/2 parts, ranked by A344296.
A035363 counts partitions of n with n/2 parts, ranked by A340387.
A056239 adds up prime indices, row sums of A112798.
A300061 ranks partitions of even numbers, with 5-smooth case A344297.
Cf. A000041, A000244, A026811, A080193, A244990, A261144, A279622.

Programs

  • Mathematica
    Select[Range[1000],PrimeOmega[#]<=Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]/2&&Max@@First/@FactorInteger[#]<=5&]

Formula

Intersection of A051037 and A344291.

A344294 5-smooth but not 3-smooth numbers k such that A056239(k) >= 2*A001222(k).

Original entry on oeis.org

5, 10, 15, 25, 30, 45, 50, 75, 90, 100, 125, 135, 150, 225, 250, 270, 300, 375, 405, 450, 500, 625, 675, 750, 810, 900, 1000, 1125, 1215, 1250, 1350, 1500, 1875, 2025, 2250, 2430, 2500, 2700, 3000, 3125, 3375, 3645, 3750, 4050, 4500, 5000, 5625, 6075, 6250
Offset: 1

Views

Author

Gus Wiseman, May 16 2021

Keywords

Comments

A number is d-smooth iff its prime divisors are all <= d.
A prime index of k is a number m such that prime(m) divides k, and the multiset of prime indices of k is row k of A112798. This row has length A001222(k) and sum A056239(k).

Examples

			The sequence of terms together with their prime indices begins:
       5: {3}           270: {1,2,2,2,3}
      10: {1,3}         300: {1,1,2,3,3}
      15: {2,3}         375: {2,3,3,3}
      25: {3,3}         405: {2,2,2,2,3}
      30: {1,2,3}       450: {1,2,2,3,3}
      45: {2,2,3}       500: {1,1,3,3,3}
      50: {1,3,3}       625: {3,3,3,3}
      75: {2,3,3}       675: {2,2,2,3,3}
      90: {1,2,2,3}     750: {1,2,3,3,3}
     100: {1,1,3,3}     810: {1,2,2,2,2,3}
     125: {3,3,3}       900: {1,1,2,2,3,3}
     135: {2,2,2,3}    1000: {1,1,1,3,3,3}
     150: {1,2,3,3}    1125: {2,2,3,3,3}
     225: {2,2,3,3}    1215: {2,2,2,2,2,3}
     250: {1,3,3,3}    1250: {1,3,3,3,3}

Crossrefs

Allowing any number of parts and sum gives A080193, counted by A069905.
The partitions with these Heinz numbers are counted by A325691.
Relaxing the smoothness conditions gives A344291, counted by A110618.
Allowing 3-smoothness gives A344293, counted by A266755.
A025065 counts partitions of n with at least n/2 parts, ranked by A344296.
A035363 counts partitions of n whose length is n/2, ranked by A340387.
A051037 lists 5-smooth numbers (complement: A279622).
A056239 adds up prime indices, row sums of A112798.
A257993 gives the least gap of the partition with Heinz number n.
A300061 lists numbers with even sum of prime indices (5-smooth: A344297).
A342050/A342051 list Heinz numbers of partitions with even/odd least gap.
Cf. A000041, A001399, A002865, A027336, A244990, A261144, A344295.

Programs

  • Mathematica
    Select[Range[1000],PrimeOmega[#]<=Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]/2&&Max@@First/@FactorInteger[#]==5&]

Formula

Intersection of A080193 and A344291.

A344413 Numbers n whose sum of prime indices A056239(n) is even and is at least twice the number of prime factors A001222(n).

Original entry on oeis.org

1, 3, 7, 9, 10, 13, 19, 21, 22, 25, 27, 28, 29, 30, 34, 37, 39, 43, 46, 49, 52, 53, 55, 57, 61, 62, 63, 66, 70, 71, 75, 76, 79, 81, 82, 84, 85, 87, 88, 89, 90, 91, 94, 100, 101, 102, 107, 111, 113, 115, 116, 117, 118, 121, 129, 130, 131, 133, 134, 136, 138
Offset: 1

Views

Author

Gus Wiseman, May 19 2021

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Also Heinz numbers of integer partitions of even numbers m with at most m/2 parts, counted by A209816 riffled with zeros, or A110618 with odd positions zeroed out.

Examples

			The sequence of terms together with their prime indices begins:
      1: {}          37: {12}        75: {2,3,3}
      3: {2}         39: {2,6}       76: {1,1,8}
      7: {4}         43: {14}        79: {22}
      9: {2,2}       46: {1,9}       81: {2,2,2,2}
     10: {1,3}       49: {4,4}       82: {1,13}
     13: {6}         52: {1,1,6}     84: {1,1,2,4}
     19: {8}         53: {16}        85: {3,7}
     21: {2,4}       55: {3,5}       87: {2,10}
     22: {1,5}       57: {2,8}       88: {1,1,1,5}
     25: {3,3}       61: {18}        89: {24}
     27: {2,2,2}     62: {1,11}      90: {1,2,2,3}
     28: {1,1,4}     63: {2,2,4}     91: {4,6}
     29: {10}        66: {1,2,5}     94: {1,15}
     30: {1,2,3}     70: {1,3,4}    100: {1,1,3,3}
     34: {1,7}       71: {20}       101: {26}
For example, 75 has 3 prime indices {2,3,3} with sum 8 >= 2*3, so 75 is in the sequence.

Links

Crossrefs

These are the Heinz numbers of partitions counted by A209816 and A110618.
A subset of A300061 (sum of prime indices is even).
The conjugate version appears to be A320924 (allowing odd weights: A322109).
The case of equality is A340387.
Allowing odd weights gives A344291.
The 5-smooth case is A344295, or A344293 allowing odd weights.
The opposite version allowing odd weights is A344296.
The conjugate opposite version allowing odd weights is A344414.
The case of equality in the conjugate case is A344415.
The conjugate opposite version is A344416, counted by A000070.
A001222 counts prime factors with multiplicity.
A027187 counts partitions of even length, ranked by A028260.
A056239 adds up prime indices, row sums of A112798.
A058696 counts partitions of even numbers, ranked by A300061.
A301987 lists numbers whose sum of prime indices equals their product.
A330950 counts partitions of n with Heinz number divisible by n.
A334201 adds up all prime indices except the greatest.
Cf. A001414, A025065, A067538, A301988, A316413, A316428, A325037, A325038, A325044, A338914, A344294, A344297.

Programs

  • Maple
    filter:= proc(n) local F,a,t;
  F:= ifactors(n)[2];
  a:= add((numtheory:-pi(t[1])-2)*t[2],t=F);
  a::even and a >= 0
end proc:
select(filter, [$1..300]); # Robert Israel, Oct 10 2024
  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],EvenQ[Total[primeMS[#]]]&&PrimeOmega[#]<=Total[primeMS[#]]/2&]

Formula

Members m of A300061 such that A056239(m) >= 2*A001222(m).
Showing 1-4 of 4 results.

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